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  "body": "\n## Definition\n\nLet $S$ be a [[semigroup|0XMEM406JM]].\nThe semigroup $S$ is an *inverse semigroup* if for every $a \\in S$ there is exactly one $b \\in S$ with\n$$aba = a \\quad \\text{and} \\quad bab = b.$$\nThis $b$ is written $a^{-1}$.\n\n## Remarks\n\nEquivalently, $S$ is [[regular|XGJKF6T7FZ]] and any two [[idempotents|0N5NJZKKN3]] of $S$ commute.\nIn an inverse semigroup, $(a^{-1})^{-1} = a$ and $(ab)^{-1} = b^{-1} a^{-1}$.\n\nExample: the symmetric inverse monoid $I_X$ on a set $X$, the set of all bijections $f \\colon A \\to B$ between subsets $A, B \\subseteq X$.\nThe product $fg$ is the composite $f \\circ g$, defined on the points $x$ with $x$ in the domain of $g$ and $g(x)$ in the domain of $f$.\nFor $f \\colon A \\to B$, the inverse bijection $f^{-1} \\colon B \\to A$ satisfies $f f^{-1} f = f$ and $f^{-1} f f^{-1} = f^{-1}$, and it is the only element that does.\nEvery group is also an inverse semigroup.\n\nNon-example: the left zero semigroup $\\set{e, f}$ with $xy = x$ for all $x, y$.\nHere $exe = e$ and $xex = x$ for both $x = e$ and $x = f$, so $e$ has two such partners and uniqueness fails.\nAccordingly, the idempotents $e$ and $f$ do not commute: $ef = e$ but $fe = f$.\n",
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